Verfügbarkeit: Outer circles

Outer circles: an introduction to hyperbolic 3-Manifolds

We live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of th...

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1. Verfasser:	Marden, Albert (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Cambridge Cambridge Univ. Press 2007
Ausgabe:	1. publ.
Schlagworte:	Three-manifolds (Topology) Hyperbolische Geometrie Dimension 3 Hyperbolische Mannigfaltigkeit
Online-Zugang:	Inhaltsverzeichnis
Zusammenfassung:	We live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of the subject of hyperbolic manifolds as a whole. The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study.
Beschreibung:	XVII, 427 S. Ill.
ISBN:	9780521839747 0521839742

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Datensatz im Suchindex

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adam_text	Contents List of Illustrations page xi Preface xiii 1 Hyperbolic space and its isometries 1 1.1 Möbius transformations 1 1.2 Hyperbolic geometry 6 1.3 The circle or sphere at infinity 11 1.4 Gaussian curvature 15 1.5 Further properties of Möbius transformations 18 1.6 Exercises and explorations 23 2 Discrete groups 49 2.1 Convergence of Möbius transformations 49 2.2 Discreteness 51 2.3 Elementary discrete groups 55 2.4 Kleinian groups 58 2.5 Quotient manifolds and orbifolds 62 2.5.1 Two fundamental algebraic theorems 68 2.6 Introduction to Riemann surfaces and their uniformization 69 2.7 Fuchsian and Schottky groups 74 2.8 Riemannian metrics and quasiconformal mappings 78 2.9 Exercises and explorations 83 3 Properties of hyperbolic manifolds 105 3.1 The Ahlfors Finiteness Theorem 105 3.2 Tubes and horoballs 106 3.3 Universal properties 108 3.4 The thick/thin decomposition of a manifold 115 3.5 Fundamental polyhedra 116 3.6 Geometric finiteness 124 3.7 Three-manifold surgery 129 3.8 Quasifuchsian groups 134 3.9 Geodesic and measured geodesic laminations 136 viii Contents 3.1 0 The convex hull of the limit set 144 3.11 The convex core 151 3.12 The compact and relative compact core 155 3.13 Rigidity 156 3.14 Exercises and explorations 161 4 Algebraic and geometric convergence 187 4.1 Algebraic convergence 187 4.2 Geometric convergence 193 4.3 Polyhedral convergence 194 4.4 The geometric limit 197 4.5 Convergence of limit sets and regions of discontinuity 200 4.6 New parabolics 203 4.7 Acylindrical manifolds 205 4.8 Dehn surgery 207 4.9 The prototypical example 208 4.10 Manifolds of finite volume 211 4.11 The Dehn surgery theorems for finite volume manifolds 212 4.12 Exercises and explorations 218 5 Deformation spaces and the ends of manifolds 239 5.1 The representation variety 239 5.2 Homotopy equivalence 244 5.3 The quasiconformal deformation space boundary . 248 5.4 The three great conjectures 250 5.5 Ends of hyperbolic manifolds 251 5.6 Tame manifolds 252 5.7 Quasifuchsian spaces 261 5.8 The quasifuchsian space boundary 265 5.9 Geometric limits at boundary points 271 5.10 Exercises and explorations 282 6 Hyperbolization 312 6.1 Hyperbolic manifolds that fiber over a circle 312 6.1.1 Automorphisms of surfaces 312 6.1.2 The Double Limit Theorem 314 6.1.3 Manifolds fibered over the circle 315 6.2 The Skinning Lemma 317 6.2.1 Hyperbolic manifolds with totally geodesic boundary 317 6.2.2 Skinning the manifold (Part II) 319 6.3 The Hyperbolization Theorem 322 6.3.1 Knots and links 325 6.4 Geometrization 327 6.5 The Orbifold Theorem 329 6.6 Exercises and Explorations 331 Com ent s ix 7 Line geometry 348 7.1 Half-rotations 348 7.2 The Lie product 349 7.3 Square roots 353 7.4 Complex distance 353 7.5 Complex distance and line geometry 355 7.6 Exercises and explorations 356 8 Right hexagons and hyperbolic trigonometry 366 8.1 Generic right hexagons 366 8.2 The sine and cosine laws for generic right hexagons 368 8.3 Degenerate right hexagons 370 8.4 Formulas for triangles, quadrilaterals, and pentagons 372 8.5 Exercises and explorations 375 Bibliography 393 Index ■ 411
adam_txt	Contents List of Illustrations page xi Preface xiii 1 Hyperbolic space and its isometries 1 1.1 Möbius transformations 1 1.2 Hyperbolic geometry 6 1.3 The circle or sphere at infinity 11 1.4 Gaussian curvature 15 1.5 Further properties of Möbius transformations 18 1.6 Exercises and explorations 23 2 Discrete groups 49 2.1 Convergence of Möbius transformations 49 2.2 Discreteness 51 2.3 Elementary discrete groups 55 2.4 Kleinian groups 58 2.5 Quotient manifolds and orbifolds 62 2.5.1 Two fundamental algebraic theorems 68 2.6 Introduction to Riemann surfaces and their uniformization 69 2.7 Fuchsian and Schottky groups 74 2.8 Riemannian metrics and quasiconformal mappings 78 2.9 Exercises and explorations 83 3 Properties of hyperbolic manifolds 105 3.1 The Ahlfors Finiteness Theorem 105 3.2 Tubes and horoballs 106 3.3 Universal properties 108 3.4 The thick/thin decomposition of a manifold 115 3.5 Fundamental polyhedra 116 3.6 Geometric finiteness 124 3.7 Three-manifold surgery 129 3.8 Quasifuchsian groups 134 3.9 Geodesic and measured geodesic laminations 136 viii Contents 3.1 0 The convex hull of the limit set 144 3.11 The convex core 151 3.12 The compact and relative compact core 155 3.13 Rigidity 156 3.14 Exercises and explorations 161 4 Algebraic and geometric convergence 187 4.1 Algebraic convergence 187 4.2 Geometric convergence 193 4.3 Polyhedral convergence 194 4.4 The geometric limit 197 4.5 Convergence of limit sets and regions of discontinuity 200 4.6 New parabolics 203 4.7 Acylindrical manifolds 205 4.8 Dehn surgery 207 4.9 The prototypical example 208 4.10 Manifolds of finite volume 211 4.11 The Dehn surgery theorems for finite volume manifolds 212 4.12 Exercises and explorations 218 5 Deformation spaces and the ends of manifolds 239 5.1 The representation variety 239 5.2 Homotopy equivalence 244 5.3 The quasiconformal deformation space boundary . 248 5.4 The three great conjectures 250 5.5 Ends of hyperbolic manifolds 251 5.6 Tame manifolds 252 5.7 Quasifuchsian spaces 261 5.8 The quasifuchsian space boundary 265 5.9 Geometric limits at boundary points 271 5.10 Exercises and explorations 282 6 Hyperbolization 312 6.1 Hyperbolic manifolds that fiber over a circle 312 6.1.1 Automorphisms of surfaces 312 6.1.2 The Double Limit Theorem 314 6.1.3 Manifolds fibered over the circle 315 6.2 The Skinning Lemma 317 6.2.1 Hyperbolic manifolds with totally geodesic boundary 317 6.2.2 Skinning the manifold (Part II) 319 6.3 The Hyperbolization Theorem 322 6.3.1 Knots and links 325 6.4 Geometrization 327 6.5 The Orbifold Theorem 329 6.6 Exercises and Explorations 331 Com ent s ix 7 Line geometry 348 7.1 Half-rotations 348 7.2 The Lie product 349 7.3 Square roots 353 7.4 Complex distance 353 7.5 Complex distance and line geometry 355 7.6 Exercises and explorations 356 8 Right hexagons and hyperbolic trigonometry 366 8.1 Generic right hexagons 366 8.2 The sine and cosine laws for generic right hexagons 368 8.3 Degenerate right hexagons 370 8.4 Formulas for triangles, quadrilaterals, and pentagons 372 8.5 Exercises and explorations 375 Bibliography 393 Index ■ 411
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spelling	Marden, Albert Verfasser (DE-588)133428737 aut Outer circles an introduction to hyperbolic 3-Manifolds A. Marden 1. publ. Cambridge Cambridge Univ. Press 2007 XVII, 427 S. Ill. txt rdacontent n rdamedia nc rdacarrier We live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of the subject of hyperbolic manifolds as a whole. The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study. Three-manifolds (Topology) Hyperbolische Geometrie (DE-588)4161041-6 gnd rswk-swf Dimension 3 (DE-588)4321722-9 gnd rswk-swf Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 gnd rswk-swf Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 s Dimension 3 (DE-588)4321722-9 s DE-604 Hyperbolische Geometrie (DE-588)4161041-6 s Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016029653&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Marden, Albert Outer circles an introduction to hyperbolic 3-Manifolds Three-manifolds (Topology) Hyperbolische Geometrie (DE-588)4161041-6 gnd Dimension 3 (DE-588)4321722-9 gnd Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 gnd
subject_GND	(DE-588)4161041-6 (DE-588)4321722-9 (DE-588)4161044-1
title	Outer circles an introduction to hyperbolic 3-Manifolds
title_auth	Outer circles an introduction to hyperbolic 3-Manifolds
title_exact_search	Outer circles an introduction to hyperbolic 3-Manifolds
title_exact_search_txtP	Outer circles an introduction to hyperbolic 3-Manifolds
title_full	Outer circles an introduction to hyperbolic 3-Manifolds A. Marden
title_fullStr	Outer circles an introduction to hyperbolic 3-Manifolds A. Marden
title_full_unstemmed	Outer circles an introduction to hyperbolic 3-Manifolds A. Marden
title_short	Outer circles
title_sort	outer circles an introduction to hyperbolic 3 manifolds
title_sub	an introduction to hyperbolic 3-Manifolds
topic	Three-manifolds (Topology) Hyperbolische Geometrie (DE-588)4161041-6 gnd Dimension 3 (DE-588)4321722-9 gnd Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 gnd
topic_facet	Three-manifolds (Topology) Hyperbolische Geometrie Dimension 3 Hyperbolische Mannigfaltigkeit
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